Fast algorithms for spectral differentiation matrices

Abstract

Recently Olver and Townsend presented a fast spectral method that relies on bases of ultraspherical polynomials to give differentiation matrices that are almost banded. The almost-banded structure allowed them to develop efficient algorithms for solving ceratin discretized systems in linear time. We show that one can also design fast algorithms for standard spectral methods because the underlying matrices, though dense, have the same rank structure as those of Olver and Townsend.